In order to breed two creatures, place a male and female creature next to each other and click on the heart icon. Sometimes the breeding will fail if one of the creatures has low fertility. If it succeeds, the female will get a swollen belly added to their model, signifying pregnancy. They will also have this in their personal menu. Females will only give birth if they are sitting on a nest.
Creatures cannot breed if they are too young (less than 3 gems), already pregnant, or tending to a baby on a nest. If the female has one day left before death, they will give birth before they die, leaving the baby unguarded unless another adult is nearby. If one of the parents has no moves left, you can still mate using the other creature's action point.
Each parent will randomly pass on one copy of their genes. The dominance of certain genes has a large role to play in what the child will look like or how it will function.
The fertility genes increase or lower the chance of breeding to succeed, and increase the possibility of twins. Each animal has two copies of the gene, and adding up each copy equals how likely it is for the breeding to work.
If a creature has high enough fertility, that creature will be able to produce twins. The twin creatures may or may not be alike, because the genes are inherited randomly. Both creatures will either receive a mutation or a gene from one or both of their parents.
When you destroy a nest with a female and twins in, the twins and female layer upon each other and cause them to mold together as the picture shows, it is even worse when you move the female away.